nLab
effective monomorphism

Idea

• effective epimorphism $\Rightarrow$ regular epimorphism $\iff$ covering

• effective monomorphism $\Rightarrow$ regular monomorphism $\iff$ embedding .

Definition

A morphism $f:X\to Y$ in a category $C$ is an effective monomorphism if

1. it has a cokernel pair, i.e. if the pushout $Y{\coprod }_{X}Y$ exists;

2. it is the equalizer of the canonical pair of morphisms $Y\overrightarrow{\to }Y{\coprod }_{X}Y$.

Such $f$ may probably also be called an embedding.

The dual concept is that of effective epimorphism. See there for more discussion.